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            <h1 style="display: none">第三章 不定积分和定积分</h1>
            
            
              <div class="markdown-body">
                
                <h2 id="不定积分"><a href="#不定积分" class="headerlink" title="不定积分"></a>不定积分</h2><h3 id="一、不定积分的概念和性质"><a href="#一、不定积分的概念和性质" class="headerlink" title="一、不定积分的概念和性质"></a>一、不定积分的概念和性质</h3><h6 id="概念"><a href="#概念" class="headerlink" title="概念"></a>概念</h6><p><img src="https://i0.hdslb.com/bfs/album/01031f180025ce3f8c7b5254388d7e693dc4636d.png" srcset="/img/loading.gif" lazyload alt="不定积分的概念和性质"></p>
<p><img src="https://i0.hdslb.com/bfs/album/cce5306c2bbfa3f27a3263b9150ba9c6a21fc05e.png" srcset="/img/loading.gif" lazyload alt="不定积分"></p>
<p><img src="https://i0.hdslb.com/bfs/album/382162d2ee27e27668dfdb9b0d18ff33816293ae.png" srcset="/img/loading.gif" lazyload alt="书写形式"></p>
<p><img src="https://i0.hdslb.com/bfs/album/993ea6d158d56d198bc998b14c988cfad389c5c0.png" srcset="/img/loading.gif" lazyload alt="注意"></p>
<h6 id="性质"><a href="#性质" class="headerlink" title="性质"></a>性质</h6><p><img src="https://i0.hdslb.com/bfs/album/4cc176f9d67b3164b9691c95462602da71953e04.png" srcset="/img/loading.gif" lazyload alt="不定积分性质"></p>
<h3 id="二、原函数练习题"><a href="#二、原函数练习题" class="headerlink" title="二、原函数练习题"></a>二、原函数练习题</h3><h6 id="例题"><a href="#例题" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/1435a8362ee5a337d758321a32196654a400e0c7.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/94859d62749cb8a9aa469ceefd15af7d5a640c4d.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/7c30819db8fce820ad0472440c716a8c650fac1c.png" srcset="/img/loading.gif" lazyload alt="例题三"></p>
<p><img src="https://i0.hdslb.com/bfs/album/32155be86a215c48cd3259446607c1d24e52f135.png" srcset="/img/loading.gif" lazyload alt="例题四"></p>
<p><img src="https://i0.hdslb.com/bfs/album/bdb26b2f1594e167d545f61417ab0dace6668736.png" srcset="/img/loading.gif" lazyload alt="例题五"></p>
<p><img src="https://i0.hdslb.com/bfs/album/3fb588dfaed1b4067861b117ddfe28337907515e.png" srcset="/img/loading.gif" lazyload alt="例题五"></p>
<h3 id="三、不定积分性质练习题"><a href="#三、不定积分性质练习题" class="headerlink" title="三、不定积分性质练习题"></a>三、不定积分性质练习题</h3><p><img src="https://i0.hdslb.com/bfs/album/a42adf778959340e7a41e4ff29357b03f319a480.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/819c77b3be9536a90c8595f9b65f2ecf001465a3.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/3011e41637e65f664b23ff7575d8bed5bd739c90.png" srcset="/img/loading.gif" lazyload alt="例题三"></p>
<h3 id="四、直接积分法练习题1"><a href="#四、直接积分法练习题1" class="headerlink" title="四、直接积分法练习题1"></a>四、直接积分法练习题1</h3><h6 id="题型"><a href="#题型" class="headerlink" title="题型"></a>题型</h6><p><img src="https://i0.hdslb.com/bfs/album/590351faf59e915a57fcc7ce389aa8add5c2ecea.png" srcset="/img/loading.gif" lazyload alt="题型"></p>
<h6 id="例题-1"><a href="#例题-1" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/36622cc9cc87bae1444a2e04b389d5785bf03153.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/f0fddcf8ae2dd70fde328c42d5c6886a38f8afbc.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/a7668f3beb04dc29ad03eaddef9bda0137064b88.png" srcset="/img/loading.gif" lazyload alt="例题三"></p>
<p><img src="https://i0.hdslb.com/bfs/album/a2a5d1838b4c88d44954a0b500f785e3d141cd78.png" srcset="/img/loading.gif" lazyload alt="例题四"></p>
<p><img src="https://i0.hdslb.com/bfs/album/fb0d97973c7e45307b7aa8a283e01542b27d8773.png" srcset="/img/loading.gif" lazyload alt="例题四"></p>
<p><img src="https://i0.hdslb.com/bfs/album/9f8e5847b5d7dfce9c32e9769badf55138848682.png" srcset="/img/loading.gif" lazyload alt="例题五"></p>
<p><img src="https://i0.hdslb.com/bfs/album/bf50af3af08ef1585a395cd81a74e48cd21d337a.png" srcset="/img/loading.gif" lazyload alt="例题六"></p>
<p><img src="https://i0.hdslb.com/bfs/album/582fd6085ae3e9723b99f3ffd3ea908e4f18aeed.png" srcset="/img/loading.gif" lazyload alt="解法二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/cffe70bd09e7fc3f28e1f81fcde8995dcac2cbdd.png" srcset="/img/loading.gif" lazyload alt="例题七"></p>
<p><img src="https://i0.hdslb.com/bfs/album/157ac1bb4d393b885030b65a70dead9fa2137b32.png" srcset="/img/loading.gif" lazyload alt="例题八"></p>
<p><img src="https://i0.hdslb.com/bfs/album/45fd6e536594c01bb46321b5908dfcdfdf3bf955.png" srcset="/img/loading.gif" lazyload alt="例题八"></p>
<h3 id="五、直接积分法练习题2"><a href="#五、直接积分法练习题2" class="headerlink" title="五、直接积分法练习题2"></a>五、直接积分法练习题2</h3><h6 id="指数型"><a href="#指数型" class="headerlink" title="指数型"></a>指数型</h6><p><img src="https://i0.hdslb.com/bfs/album/65dab6ba5259f8b73a74b3f193d36e675c80f682.png" srcset="/img/loading.gif" lazyload alt="公式"></p>
<h6 id="例题-2"><a href="#例题-2" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/e4fdba42e0951b341af0f41a3eeb39ef2fccb83a.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/24223a53472266530cd99919833e8c2cc076a5e1.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/6ad562e75d894ba80a94d5c709d5c9d96dabb4c6.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<h3 id="六、直接积分法-三角函数型3"><a href="#六、直接积分法-三角函数型3" class="headerlink" title="六、直接积分法-三角函数型3"></a>六、直接积分法-三角函数型3</h3><h6 id="公式"><a href="#公式" class="headerlink" title="公式"></a>公式</h6><p><img src="https://i0.hdslb.com/bfs/album/ad3a879027eb13423e32d974e7701b21f8aff493.png" srcset="/img/loading.gif" lazyload alt="公式"></p>
<h6 id="例题-3"><a href="#例题-3" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/6294d2648f676d4f42d0a17b235a9e2689a8eb3c.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/94425ec8a48da13789bc9210624d787df700fd5e.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/bcc657fbc1162089ef1440780d3a72e999ec162a.png" srcset="/img/loading.gif" lazyload alt="例题三"></p>
<p><img src="https://i0.hdslb.com/bfs/album/16b730bffed271cb8599ffdea1030cf50b0026c4.png" srcset="/img/loading.gif" lazyload alt="例题三"></p>
<p><img src="https://i0.hdslb.com/bfs/album/7a2b3dd36c9eef64a91168a8e8a7d21ef3eb899d.png" srcset="/img/loading.gif" lazyload alt="例题四"></p>
<h3 id="七、凑微分法1"><a href="#七、凑微分法1" class="headerlink" title="七、凑微分法1"></a>七、凑微分法1</h3><h6 id="思想"><a href="#思想" class="headerlink" title="思想"></a>思想</h6><p><img src="https://i0.hdslb.com/bfs/album/37f9fd6d13bcde0e40b608b69fc5d9c5e5983d5d.png" srcset="/img/loading.gif" lazyload alt="基本思想"></p>
<h6 id="题型和解题步骤"><a href="#题型和解题步骤" class="headerlink" title="题型和解题步骤"></a>题型和解题步骤</h6><p><img src="https://i0.hdslb.com/bfs/album/d185060ea5609f29a698bb452bbfe60723b44897.png" srcset="/img/loading.gif" lazyload alt="题型和解题步骤"></p>
<p><img src="https://i0.hdslb.com/bfs/album/dd9eeb14bbdfeadc5eb91587a7b7eb884b384b0a.png" srcset="/img/loading.gif" lazyload alt="题型和解题步骤"></p>
<h6 id="真题"><a href="#真题" class="headerlink" title="真题"></a>真题</h6><p>2012</p>
<p><img src="https://i0.hdslb.com/bfs/album/8b5898ed636f226ee53d6a9b6bf1e3e58f44af19.png" srcset="/img/loading.gif" lazyload alt="2012-6"></p>
<p><img src="https://i0.hdslb.com/bfs/album/aa16b5e10026cab8df303c9c375c4cb3fa222053.png" srcset="/img/loading.gif" lazyload alt="2012-6"></p>
<p>2020</p>
<p><img src="https://i0.hdslb.com/bfs/album/cfc0e68b852c3ebb75cbea5a47896a84023205a1.png" srcset="/img/loading.gif" lazyload alt="2020"></p>
<p><img src="https://i0.hdslb.com/bfs/album/d338a7d8ebb6a83396579675545102b168ab168d.png" srcset="/img/loading.gif" lazyload alt="2020"></p>
<p>2020</p>
<p><img src="https://i0.hdslb.com/bfs/album/86fb72e04bdd75b5591d8e18c8c28cd3e6da0a14.png" srcset="/img/loading.gif" lazyload alt="2020"></p>
<p><img src="https://i0.hdslb.com/bfs/album/22d4d32297e33d3f01f48f260e8f4f5f597cb6e4.png" srcset="/img/loading.gif" lazyload alt="2020"></p>
<p>2017</p>
<p><img src="https://i0.hdslb.com/bfs/album/a2b04f40e34ac13caeb04fe5b34448fe0a3ea250.png" srcset="/img/loading.gif" lazyload alt="2017"></p>
<p><img src="https://i0.hdslb.com/bfs/album/4ffd3f08fb2075b93c1ee290ad3ac798a7578aba.png" srcset="/img/loading.gif" lazyload alt="2017"></p>
<h3 id="八、凑微分法2"><a href="#八、凑微分法2" class="headerlink" title="八、凑微分法2"></a>八、凑微分法2</h3><h6 id="题型-1"><a href="#题型-1" class="headerlink" title="题型"></a>题型</h6><p><img src="https://i0.hdslb.com/bfs/album/8e95a7abf2841af593c0ce2718136ac6ba662013.png" srcset="/img/loading.gif" lazyload alt="题型"></p>
<h6 id="真题-1"><a href="#真题-1" class="headerlink" title="真题"></a>真题</h6><p>2018</p>
<p><img src="https://i0.hdslb.com/bfs/album/f6ed35d54f20f76874c9e9b9d3d00307cd1618a6.png" srcset="/img/loading.gif" lazyload alt="2018"></p>
<p><img src="https://i0.hdslb.com/bfs/album/61b7513e923c88b25c022e5f1553903a2cd8e601.png" srcset="/img/loading.gif" lazyload alt="2018"></p>
<h3 id="九、凑微分法3"><a href="#九、凑微分法3" class="headerlink" title="九、凑微分法3"></a>九、凑微分法3</h3><h6 id="题型-2"><a href="#题型-2" class="headerlink" title="题型"></a>题型</h6><p><img src="https://i0.hdslb.com/bfs/album/d1f5b5cfa19cc3e986d18b90378643689d9c2778.png" srcset="/img/loading.gif" lazyload alt="题型"></p>
<h6 id="例题-4"><a href="#例题-4" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/743e6667fd4bcba5d953f5136b725505f2789cb6.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/3e7fc6f92c2740ef130d0ed06b913b4ad9504153.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/c4d6d1154d632701bacaf9861bbb1afe2197c4d1.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<h6 id="真题-2"><a href="#真题-2" class="headerlink" title="真题"></a>真题</h6><p>2015</p>
<p><img src="https://i0.hdslb.com/bfs/album/2166cb994ac63f78cbb858fe4b3f806c20a300a4.png" srcset="/img/loading.gif" lazyload alt="2015"></p>
<p><img src="https://i0.hdslb.com/bfs/album/5281a9a1161ce24ad717b84d4eb982f957ee43ad.png" srcset="/img/loading.gif" lazyload alt="2015"></p>
<p><img src="https://i0.hdslb.com/bfs/album/b5cb6a8808f63fe4b2d3ef69df137967e388838a.png" srcset="/img/loading.gif" lazyload alt="2015"></p>
<h3 id="十、凑微分法4"><a href="#十、凑微分法4" class="headerlink" title="十、凑微分法4"></a>十、凑微分法4</h3><h6 id="题型-3"><a href="#题型-3" class="headerlink" title="题型"></a>题型</h6><p><img src="https://i0.hdslb.com/bfs/album/3c6c82b41ed51764dfc10bf07a0f0412150a6c5f.png" srcset="/img/loading.gif" lazyload alt="题型"></p>
<h6 id="例题-5"><a href="#例题-5" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/28175ea432b8a54b4f6edbee93961a49b80b67b3.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/01419d440a468ea610468fd0b4923bc94c43822d.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/33ef46d67872c21aa5cbd0931614cbb4dea1a932.png" srcset="/img/loading.gif" lazyload alt="例题三"></p>
<h3 id="十一、无理根式换元法"><a href="#十一、无理根式换元法" class="headerlink" title="十一、无理根式换元法"></a>十一、无理根式换元法</h3><h6 id="解题"><a href="#解题" class="headerlink" title="解题"></a>解题</h6><p><img src="https://i0.hdslb.com/bfs/album/2292df01b88f229ee3f9e93324d5ab948f254d93.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-6"><a href="#例题-6" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/00e1dc2e05bd9b943a7e9c0aa221ad7fce17a5d4.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/6baac540cf1a4b6185276619af4def086e83a73f.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/b038980959bcff6db2d7a7c7d242e7a7b55696ce.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/6acff36b12892b4d6149042dc78d0dae08a48986.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/2b0b64f364308f0f9f26aa1e4a62d95cd89bff71.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/7c1ed6c59eeb97abd21b4cfa6521b6d1742ac86a.png" srcset="/img/loading.gif" lazyload alt="例题三"></p>
<p><img src="https://i0.hdslb.com/bfs/album/fd4582d038e89828578eb098511277712b1feae0.png" srcset="/img/loading.gif" lazyload alt="例题三"></p>
<h6 id="题型-4"><a href="#题型-4" class="headerlink" title="题型"></a>题型</h6><p><img src="https://i0.hdslb.com/bfs/album/6fe2c16aedcc9089b551d945ca4b7c7f623ad675.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-7"><a href="#例题-7" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/4b48f2ec0458af5e152d656e774164dac2c0f555.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/2b98d78111ca46dc4eefe501eace398d04615282.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/65fcc608e3f48942b0eb460f2094e49ff633eb25.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="十二、三角代换"><a href="#十二、三角代换" class="headerlink" title="十二、三角代换"></a>十二、三角代换</h3><h6 id="公式-1"><a href="#公式-1" class="headerlink" title="公式"></a>公式</h6><p><img src="https://i0.hdslb.com/bfs/album/d360df8e958ddb592daf12c5a28db3193d0d54c2.png" srcset="/img/loading.gif" lazyload></p>
<img src="https://i0.hdslb.com/bfs/album/1b91d4c9711bc5ed71507e023c4325994130b89d.png" srcset="/img/loading.gif" lazyload alt="image-20220928232447502" style="zoom: 200%;" />

<p><img src="https://i0.hdslb.com/bfs/album/58cec705ae8f0b9adcc94b51931ce1f4af42efd5.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="步骤"><a href="#步骤" class="headerlink" title="步骤"></a>步骤</h6><p><img src="https://i0.hdslb.com/bfs/album/af7b3f496b8a4f08052866879d47f3e04211e3e3.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="三角关系"><a href="#三角关系" class="headerlink" title="三角关系"></a>三角关系</h6><p><img src="https://i0.hdslb.com/bfs/album/c60052eea2910dc240ad565f0ea844c8262d062c.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-8"><a href="#例题-8" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/435cfaa6341cd729f932894eb48faa4b5ddf84ab.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/976677f885da8ce08a3ffa1249a7071e107171f3.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="十三、分部积分"><a href="#十三、分部积分" class="headerlink" title="十三、分部积分"></a>十三、分部积分</h3><h6 id="公式-2"><a href="#公式-2" class="headerlink" title="公式"></a>公式</h6><p><img src="https://i0.hdslb.com/bfs/album/b767cacbb1d50644190e8af938ae7f5f9c89f14a.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/86966c495e849a6b98666e1933adb7985b625619.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="使用条件"><a href="#使用条件" class="headerlink" title="使用条件"></a>使用条件</h6><p><img src="https://i0.hdslb.com/bfs/album/92aa0be9e61a1dc1ffd30feed403a96f34455ecb.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="类型一"><a href="#类型一" class="headerlink" title="类型一"></a>类型一</h6><p><strong>积分式子中，只有一个函数（反三角，对数）&#x3D;&gt;直接套分部积分公式</strong></p>
<p><img src="https://i0.hdslb.com/bfs/album/1a349892d0021c92da71fded4c3a77b3eaf1dbff.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/917f970c23f907503269085f127e1daec365fedf.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<h6 id="类型二"><a href="#类型二" class="headerlink" title="类型二"></a>类型二</h6><p><strong>积分式子中 两个不同类型函数相乘</strong></p>
<p><img src="https://i0.hdslb.com/bfs/album/8f21b6c596dc961aee945f088320b5c7aba15fd0.png" srcset="/img/loading.gif" lazyload alt="2019"></p>
<p><img src="https://i0.hdslb.com/bfs/album/595c8045f82f4dc297168e35fff27448f4865c8f.png" srcset="/img/loading.gif" lazyload alt="解法一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/3aed44dcac3e2794e9dae09b72b85d222a12c9bb.png" srcset="/img/loading.gif" lazyload alt="解法二"></p>
<h6 id="类型三"><a href="#类型三" class="headerlink" title="类型三"></a>类型三</h6><p><strong>积分式子、ex,cosx,sinx,乘积形式</strong></p>
<p><strong>分部积分（循环）；会出现两次循环；用两次分部</strong></p>
<p><img src="https://i0.hdslb.com/bfs/album/ea0824c246f89eedb4356a800d6e6cbdd6043f47.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/a4997d9c2fa9912ade00d6c1cab8c9ba36c12226.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="类型四"><a href="#类型四" class="headerlink" title="类型四"></a>类型四</h6><p><strong>积分式子中含f(x)’ f’’(x) 分部优先</strong></p>
<p><img src="https://i0.hdslb.com/bfs/album/2b1ae58a1574b530d549583bbfcc42103c51b0b0.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/524e7fabf5de3ca907457ff88c253c34b7d91755.png" srcset="/img/loading.gif" lazyload></p>
<h2 id="定积分"><a href="#定积分" class="headerlink" title="定积分"></a>定积分</h2><h3 id="十四、定积分概念"><a href="#十四、定积分概念" class="headerlink" title="十四、定积分概念"></a>十四、定积分概念</h3><p><img src="https://i0.hdslb.com/bfs/album/a67d82964497763da9df6124e178fc0c0209c69a.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="十五、定积分几何意义"><a href="#十五、定积分几何意义" class="headerlink" title="十五、定积分几何意义"></a>十五、定积分几何意义</h3><p><img src="https://i0.hdslb.com/bfs/album/bb9ff229cf9e39427de79f3194d21755f97c0358.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="十六、定积分比较大小"><a href="#十六、定积分比较大小" class="headerlink" title="十六、定积分比较大小"></a>十六、定积分比较大小</h3><h6 id="比较大小"><a href="#比较大小" class="headerlink" title="比较大小"></a>比较大小</h6><p><img src="https://i0.hdslb.com/bfs/album/b17235c1443f97e350928e6b5b5b0184e49d54ae.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-9"><a href="#例题-9" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/430eaa01e011824a0a17da352a486570a629c354.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/5d594b08bef0ea9a5859032296d546428cdbcdd1.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/c59df76a6265c171f099d6dc74761526cf41e63a.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="十七、定积分性质"><a href="#十七、定积分性质" class="headerlink" title="十七、定积分性质"></a>十七、定积分性质</h3><h6 id="公式-3"><a href="#公式-3" class="headerlink" title="公式"></a>公式</h6><p><img src="https://i0.hdslb.com/bfs/album/e620583afbc77f44b76f1345c54fbe531736b319.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-10"><a href="#例题-10" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/4f13979cb235029efeddee0308cbd18f83782eca.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="十八、定积分的计算公式"><a href="#十八、定积分的计算公式" class="headerlink" title="十八、定积分的计算公式"></a>十八、定积分的计算公式</h3><h6 id="公式-4"><a href="#公式-4" class="headerlink" title="公式"></a>公式</h6><p><img src="https://i0.hdslb.com/bfs/album/4d80421fe4ddfa85833c5b072b27a7a017ddf3c4.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-11"><a href="#例题-11" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/76e4e391c162abbb9d61039860ddec62e0da939a.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/66ab5245633195444002da0ee970b952b96ab63e.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/5d2e07f0fbf0bf4f508e5ec83c051c4ffb3b9552.png" srcset="/img/loading.gif" lazyload alt="例题三"></p>
<h6 id="真题-3"><a href="#真题-3" class="headerlink" title="真题"></a>真题</h6><p><img src="https://i0.hdslb.com/bfs/album/67ebc9a8be0b76092280fc1aed22316007ef7442.png" srcset="/img/loading.gif" lazyload alt="2015"></p>
<h3 id="十九、定积分的四大运算技巧"><a href="#十九、定积分的四大运算技巧" class="headerlink" title="十九、定积分的四大运算技巧"></a>十九、定积分的四大运算技巧</h3><h6 id="偶倍奇零"><a href="#偶倍奇零" class="headerlink" title="偶倍奇零"></a>偶倍奇零</h6><p><img src="https://i0.hdslb.com/bfs/album/60987eee55a4beafeed735ca822759d3a8cc44ab.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/5042c5d6550fd63fa7400e1dedb4b1ff1e72b255.png" srcset="/img/loading.gif" lazyload alt="偶函数"></p>
<p><img src="https://i0.hdslb.com/bfs/album/f53e3c03cee5e5e17540664ccf2bb63597d782e8.png" srcset="/img/loading.gif" lazyload alt="奇函数"></p>
<h3 id="二十、定积分偶倍奇零例题"><a href="#二十、定积分偶倍奇零例题" class="headerlink" title="二十、定积分偶倍奇零例题"></a>二十、定积分偶倍奇零例题</h3><h6 id="回顾"><a href="#回顾" class="headerlink" title="回顾"></a>回顾</h6><p><img src="https://i0.hdslb.com/bfs/album/d977efa1fbde72b7bbe09fe311a7fd838e07e56f.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-4"><a href="#真题-4" class="headerlink" title="真题"></a>真题</h6><p><img src="https://i0.hdslb.com/bfs/album/7192905dd5ba54a9f57feb46fd8fbd2bbd7f49fa.png" srcset="/img/loading.gif" lazyload alt="2018"></p>
<p><img src="https://i0.hdslb.com/bfs/album/ba4b08f495cb56e2d6de62fcc0736b5f2ac9dd0c.png" srcset="/img/loading.gif" lazyload alt="2019"></p>
<p><img src="https://i0.hdslb.com/bfs/album/86a573cc75b1d1273750797756fd295e3fc208ec.png" srcset="/img/loading.gif" lazyload alt="江苏"></p>
<p><img src="https://i0.hdslb.com/bfs/album/7fe1d46c9c9dbe45180e175d09a1ed6a966cedd7.png" srcset="/img/loading.gif" lazyload alt="2019"></p>
<p><img src="https://i0.hdslb.com/bfs/album/b75f4a34c6f64715caa8a028c538b43242cda8af.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="二十一、定积分点火公式"><a href="#二十一、定积分点火公式" class="headerlink" title="二十一、定积分点火公式"></a>二十一、定积分点火公式</h3><h6 id="公式-5"><a href="#公式-5" class="headerlink" title="公式"></a>公式</h6><p><img src="https://i0.hdslb.com/bfs/album/f80964f96ac4538bf14cde31e4a15624299fef36.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-12"><a href="#例题-12" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/46969066f4d239207848c31c4df85f1e335d793d.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/b8348f2c1b1b0677015e61d829f6d05c2cb208c6.png" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<h6 id="真题-5"><a href="#真题-5" class="headerlink" title="真题"></a>真题</h6><p><img src="https://i0.hdslb.com/bfs/album/fb0d08d2162d086a9590b1db5e203b92f750ee7a.png" srcset="/img/loading.gif" lazyload alt="2013"></p>
<h3 id="二十二、定积分求圆面积"><a href="#二十二、定积分求圆面积" class="headerlink" title="二十二、定积分求圆面积"></a>二十二、定积分求圆面积</h3><h6 id="公式-6"><a href="#公式-6" class="headerlink" title="公式"></a>公式</h6><p><img src="https://i0.hdslb.com/bfs/album/12bc77274c89874a809d0993b415a23bebb93bfb.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-6"><a href="#真题-6" class="headerlink" title="真题"></a>真题</h6><p>2013</p>
<p><img src="https://i0.hdslb.com/bfs/album/e58748f6d485b76bf542aeb95547d7fa8ecd67ce.png" srcset="/img/loading.gif" lazyload alt="2013"></p>
<p>2019</p>
<p><img src="https://i0.hdslb.com/bfs/album/2907d5f264378145e5ed5a34af32abcd77800e1f.png" srcset="/img/loading.gif" lazyload alt="image-20220930225245595"></p>
<p><img src="https://i0.hdslb.com/bfs/album/ffef92c8f184a4f7dd0ca63a0a769bbdd7a104e3.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="二十三、定积分的换元法"><a href="#二十三、定积分的换元法" class="headerlink" title="二十三、定积分的换元法"></a>二十三、定积分的换元法</h3><h6 id="类型"><a href="#类型" class="headerlink" title="类型"></a>类型</h6><p><img src="https://i0.hdslb.com/bfs/album/a4bf71075f60b67a60d9bc13dc7a7f126eeca88c.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-13"><a href="#例题-13" class="headerlink" title="例题"></a>例题</h6><p>例题一</p>
<p><img src="https://i0.hdslb.com/bfs/album/6f18d9aa155b85a8c2fd29a1b72beb2949a7f0cb.png" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/4e1c597fde1b0b3c9a55add9a29ba53ff6b74f96.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-7"><a href="#真题-7" class="headerlink" title="真题"></a>真题</h6><p>2019</p>
<p><img src="https://i0.hdslb.com/bfs/album/98e7a09fa5a098831bec156371a06379019306d0.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/423c69abd9c7e4ec333d89fe7e5dce6d87fabb30.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="二十四、分段函数定积分"><a href="#二十四、分段函数定积分" class="headerlink" title="二十四、分段函数定积分"></a>二十四、分段函数定积分</h3><h6 id="分段形式"><a href="#分段形式" class="headerlink" title="分段形式"></a>分段形式</h6><p><img src="https://i0.hdslb.com/bfs/album/782f23e69a22114790787417d285e3a4a64b4975.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-8"><a href="#真题-8" class="headerlink" title="真题"></a>真题</h6><p>2017</p>
<p><img src="https://i0.hdslb.com/bfs/album/c596047b777945cdeb2447ca046bc9d07eb1a013.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/a34077d67ba18cfbd6cead575d883309099881a8.png" srcset="/img/loading.gif" lazyload></p>
<p>2013</p>
<p><img src="https://i0.hdslb.com/bfs/album/7fa5fb9c3510fbc6c950e2b743c771ef43c4f231.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/86398992931560fdc6e6b40d6bc89bc890f5b6c1.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="二十五、定积分的等式证明"><a href="#二十五、定积分的等式证明" class="headerlink" title="二十五、定积分的等式证明"></a>二十五、定积分的等式证明</h3><h6 id="积分区间再现公式"><a href="#积分区间再现公式" class="headerlink" title="积分区间再现公式"></a>积分区间再现公式</h6><p><img src="https://i0.hdslb.com/bfs/album/f90caf57b6a344c8c5f6e313e3dbec16dfae4be8.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="使用条件-1"><a href="#使用条件-1" class="headerlink" title="使用条件"></a>使用条件</h6><p><img src="https://i0.hdslb.com/bfs/album/a72431d81c50f02f1f42692061c8b3bb1a069c2f.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-14"><a href="#例题-14" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/9cd94441369dfc1a7cb4b5846895462f1a731826.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="二十六、定积分等式证明例题"><a href="#二十六、定积分等式证明例题" class="headerlink" title="二十六、定积分等式证明例题"></a>二十六、定积分等式证明例题</h3><h6 id="真题-9"><a href="#真题-9" class="headerlink" title="真题"></a>真题</h6><p>2018</p>
<p><img src="https://i0.hdslb.com/bfs/album/203272d4115f568a6afc646b79b5a2b8aadfda26.png" srcset="/img/loading.gif" lazyload></p>
<p>2014</p>
<p><img src="https://i0.hdslb.com/bfs/album/11cc1244b5f48827bdcadc799e160a21fb7a1457.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/664113c2978da4dda7d89dde23372cedfbceae59.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-15"><a href="#例题-15" class="headerlink" title="例题"></a>例题</h6><p>例题一</p>
<p><img src="https://i0.hdslb.com/bfs/album/f4d15062ca5647e37103bf145b8965ab75421dc6.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/54c30fd9e764faf6f1d02b011e5e0ed347175b19.png" srcset="/img/loading.gif" lazyload></p>
<p>例题二</p>
<p><img src="https://i0.hdslb.com/bfs/album/bef38524eee9b8141878962f0aa59dd88df1cc34.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/08d53dae62f24751ad58a653e7be33dc9cc6c75f.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/7033cfc8b1e38da8170c58ed794595caa5d8e585.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="二十七、定积分的几何应用"><a href="#二十七、定积分的几何应用" class="headerlink" title="二十七、定积分的几何应用"></a>二十七、定积分的几何应用</h3><h6 id="回顾-1"><a href="#回顾-1" class="headerlink" title="回顾"></a>回顾</h6><p><img src="https://i0.hdslb.com/bfs/album/44b80703d41108ae5d8210f556a3be0456de3298.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="求平面图形面积"><a href="#求平面图形面积" class="headerlink" title="求平面图形面积"></a>求平面图形面积</h6><p>X型图</p>
<p><img src="https://i0.hdslb.com/bfs/album/5a87451dff01f33db564f9fd3055d5971d40772c.png" srcset="/img/loading.gif" lazyload></p>
<p>Y型图</p>
<p><img src="https://i0.hdslb.com/bfs/album/0e97aa6e6e16b3b3908005a2310abd5f80e9ac83.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="二十八、定积分几何应用例题"><a href="#二十八、定积分几何应用例题" class="headerlink" title="二十八、定积分几何应用例题"></a>二十八、定积分几何应用例题</h3><h6 id="思路"><a href="#思路" class="headerlink" title="思路"></a>思路</h6><p><img src="https://i0.hdslb.com/bfs/album/1e83a55054178065f10e82e7a55ce69c0cc0caab.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-16"><a href="#例题-16" class="headerlink" title="例题"></a>例题</h6><p>例题一</p>
<p><img src="https://i0.hdslb.com/bfs/album/d16a8e3ad35a3576630e285aaecaed68bfa3cfff.png" srcset="/img/loading.gif" lazyload></p>
<p>例题二</p>
<p><img src="https://i0.hdslb.com/bfs/album/a279481b4e356323fab49de8e9b2b8e9d9767338.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-10"><a href="#真题-10" class="headerlink" title="真题"></a>真题</h6><p>2010</p>
<p><img src="https://i0.hdslb.com/bfs/album/93651140d93571c477c262726d8fdcf60bafdced.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/f41734466a8947c3938e5ae20d016c2287a7c5d0.png" srcset="/img/loading.gif" lazyload></p>
<h3 id="二十九、定积分求旋转体积"><a href="#二十九、定积分求旋转体积" class="headerlink" title="二十九、定积分求旋转体积"></a>二十九、定积分求旋转体积</h3><h6 id="X型图绕X轴旋转"><a href="#X型图绕X轴旋转" class="headerlink" title="X型图绕X轴旋转"></a>X型图绕X轴旋转</h6><p><img src="https://i0.hdslb.com/bfs/album/833f3ea4220e4788ca88f4d3d50eec5236711bc6.png" srcset="/img/loading.gif" lazyload alt="image-20221001014150644"></p>
<p><img src="https://i0.hdslb.com/bfs/album/eb607a98f8d7a2be4a2fa56bf2e854435c711ae5.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="Y型图绕Y轴旋转"><a href="#Y型图绕Y轴旋转" class="headerlink" title="Y型图绕Y轴旋转"></a>Y型图绕Y轴旋转</h6><p><img src="https://i0.hdslb.com/bfs/album/49485c2baac4b4d06c628e094f67d6706e1a7212.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="X型图绕Y轴"><a href="#X型图绕Y轴" class="headerlink" title="X型图绕Y轴"></a>X型图绕Y轴</h6><p><img src="https://i0.hdslb.com/bfs/album/514330383dcdbd306f203d17bad77807ea4eddd5.png" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-11"><a href="#真题-11" class="headerlink" title="真题"></a>真题</h6><p>2015</p>
<p><img src="https://i0.hdslb.com/bfs/album/c8ac2bc2741d4d0d0501a08bd07f56ac63367a9d.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/e2730f36f378387b894fceb275a465df36fbf467.png" srcset="/img/loading.gif" lazyload></p>
<p>2019</p>
<p><img src="https://i0.hdslb.com/bfs/album/01089db5638471e3fa01d448cc0c24a0d92e9243.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/da816ec2ddc91299536650d7fbb85b182dab2b56.png" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/99bdc55be75885979e7a4faad9e3494037dd71e4.png" srcset="/img/loading.gif" lazyload></p>

                
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